Stability optimization of beams conveying fluid or carrying other axially moving materials

Optimal designs, and sensitivities of such designs, are calculated for transversely vibrating structures carrying an axially moving material. The structures studied consist of piecewise uniform and initially straight beam elements conveying a piecewise constant-speed plug flow of material along their deflected axes. The beams can be supported by a distributed Winkler-type ambient medium. Viscous damping in the beam material and the ambient medium is considered. Large static axial loads may act on the beam and on the moving material. The beams are modelled with a generalized second-order Rayleigh-Timoshenko theory including the Euler-Bernoulli theory as a special case. The structures investigated may also contain taut strings and rigid bodies. Considering a given subcritical material speed, the aim of the present study is to modify the initial design of a given beam structure in such a way that the transient vibrational motion following a transverse disturbance will die out as quickly as possible. To this end, complex eigenfrequencies pertaining to transverse vibration are calculated, and design parameters are changed so as to maximally raise the imaginary part of that eigenfrequency which has the smallest such part. In one of the examples, the objective is to maximally raise the product of the real and imaginary parts of the eigenfrequency which has the smallest such product.